To characterize the connectivity patterns of \(\mathbf {G}_N\) , we first calculate the subgraph density for \(p\)  representative subgraph configurations \(\mathbf {F} = \lbrace F_1, \ldots , F_p\rbrace \) . The measurement for \(G_i\)  is stored as the \(p\) -dimensional vector of subgraph densities \(\widetilde{\mathbf {S}}_{\mathbf {F}}(G_i) = (\widetilde{S}_{F_1}(G_i), \ldots , \widetilde{S}_{F_p}(G_i))^T\) . If we view these densities as sufficient statistics for each \(G_i\) , then one can readily characterize \(\mathbf {G}_N\)  using first-order summaries of these vectors. [1]} analyzed the properties of the mean subgraph density and developed two-sample hypothesis tests for comparing the means of two network samples. There are, however, several limitations to focusing our analysis on first-order summaries of \(\widetilde{\mathbf {S}}_{\mathbf {F}}\) . First, subgraph counts (and hence densities) are often highly correlated. Not only does this discourage stability of the variance of the sample mean, but it also confiscates the marginal effect of each subgraph configuration on the network sample. Second, when the number of subgraph configurations \(p\)  is large, it becomes challenging to visualize and explore the relationships of these subgraph densities with observations in the sample. Pairwise plots of the subgraph densities would require on the order of \(p^2\)  plots, which is infeasible even for modest \(p\) .
